1 Espaces vectoriels normés

ER
E p E
(i)λ x E, p (λx) = |λ|p(x)
(ii)x, y E, p (x+y)≤p(x) + p(y).
(ii)
p E.
p(0) = 0
x E, p (x)≥0
p−1{0}={x∈E|p(x) = 0}E
x, y E, |p(x)−p(y)| ≤ p(x−y).
x E,
p(0) = p(0 ·x) = 0 ·p(x) = 0
0 = p(x−x)≤2p(x).
p−1{0}0.(i) (ii)
E.
½p(x) = p(x−y+y)≤p(x−y) + p(y)
p(y) = p(y−x+x)≤p(x−y) + p(x)

4.
p−1{0}p.
E{0}.
E x 7→ kxk.
ϕ E. x 7→ |ϕ(x)|
E E
1.
ϕIm (ϕ) = R, ϕ
ER.
p E H.
E
H, x E H
kxk=p(x).
k·k p
∀y∈H, 0≤ |p(x+y)−p(y)| ≤ p(y) = 0.
kxk= 0 x= 0
ERn≥1B= (e1,··· , en)
E.
∀x=
n
P
i=1
xiei∈E, 
kxk∞= max
1≤i≤n|xi|,
kxk1=
n
P
i=1
|xi|.
E.
k=∞k= 1 x7→ kxkk
E.
ERϕ: (x, y)7→ hx|yi
E x E,
p(x) = pϕ(x, x).
∀(x, y)∈E2,|ϕ(x, y)| ≤ p(x)p(y)

p E.
x, y E Q
∀t∈R, Q (t) = p2(x+ty) = p2(y)t2+ 2tϕ (x, y) + p2(x).
ϕ Q (t)≥0t.
p(y)=0, Q ϕ (x, y)=0.
|ϕ(x, y)|=p(x)p(y)..
p(y)6= 0, Q 2
ϕ2(x, y)−p2(x)p2(y)≤0,
|ϕ(x, y)| ≤ p(x)p(y).
x, y E
p2(x+y) = p2(y)+2ϕ(x, y) + p2(x),
ϕ(x, y)≤ |ϕ(x, y)| ≤ p(x)p(y),
p2(x+y)≤(p(x) + p(y))2.
p. p (λx) = |λ|p(x)
ϕ p
ERn≥1B= (e1,··· , en)
E.
∀x=
n
P
i=1
xiei∈E, kxk2=rn
P
i=1
|xi|2
E.
p q 1
p+1 q= 1.
ln,
∀λ∈[0,1] ,∀u∈R+,∀v∈R+, uλv1−λ≤λu + (1 −λ)v.
x y Rn.(1.1)
λ=1
p, ui=|xi|p
n
P
j=1
|xj|p
, vi=|yi|q
n
P
j=1
|yj|q
,
x y
¯¯¯¯¯
n
X
i=1
xiyi¯¯¯¯¯ ≤Ãn
X
i=1
|xi|p!1
pÃn
X
i=1
|yi|q!1
q
.

x, y Rn
n
X
i=1
|xi| |xi+yi|p−1≤Ãn
X
i=1
|xi|p!1
pÃn
X
i=1
|xi+yi|p!1
q
.
p≥1
x7→ kxkp=Ãn
X
i=1
|xi|p!1
p
Rn.
ln ln00 (t) = −1
t2<0t > 0.
]0,+∞[, λ ∈[0,1]
u, v
ln (λu + (1 −λ)v)≥λln (u) + (1 −λ) ln (v),
ln (λu + (1 −λ)v)≥ln ¡uλv1−λ¢,exp
]0,+∞[, uλv1−λ≤λu + (1 −λ)v.
u= 0 v= 0.
x, y Rn
kxkp=Ãn
X
i=1
|xi|p!1
p
kykq=Ãn
X
i=1
|yi|q!1
q
.
λ=1
p(1.1) ,1−λ=1 q
∀u≥0,∀v≥0, u1
pv1 q≤1
pu+1 qv.
x y Rn, i 1
n
ui=|xi|p
n
P
j=1
|xj|p
=|xi|p
kxkp
p
, vi=|yi|q
n
P
j=1
|yj|q
=|yi|q
kykq
q
,
|xi|
kxkp
|yi|
kykq
≤1
p
|xi|p
kxkp
p
+1
q
|yi|q
kykq
q
.
1
kxkpkykq
n
X
i=1
|xi| |yi| ≤ 1
p
n
X
i=1
|xi|p
kxkp
p
+1
q
n
X
i=1
|yi|q
kykq
q
.
n
P
i=1
|xi|p
kxkp
p
=
n
P
i=1
|yi|q
kykq
q
= 1 1
p+1 q= 1
¯¯¯¯¯
n
X
i=1
xiyi¯¯¯¯¯ ≤
n
X
i=1
|xi| |yi| ≤ kxkpkykq,
x= 0 y= 0.

hx|yi
Rn,
|hx|yi| ≤ kxkpkykq.
p=q= 2
p= 1 q= +∞.
x, y Rn
n
X
i=1
|xi| |xi+yi|p−1≤Ãn
X
i=1
|xi|p!1
pÃn
X
i=1
|xi+yi|(p−1)q!1
q
,
(p−1) q=p
n
X
i=1
|xi| |xi+yi|p−1≤Ãn
X
i=1
|xi|p!1
pÃn
X
i=1
|xi+yi|p!1
q
.
x7→ kxk1Rn. p > 1
q1
p+1 q= 1. x, y Rn
n
X
i=1
|xi+yi|p≤
n
X
i=1
|xi+yi|p−1|xi|+
n
X
i=1
|xi+yi|p−1|yi|.
³kx+ykp´p≤³kxkp+kykp´³kx+ykp´p
q,
p−p
q= 1
kx+ykp≤ kxkp+kykp.
k·kp.
xRnlim
p→+∞kxkp=kxk∞.
xRni1n|xi|p≤ kxkp
∞.
kxkp≤n1
pkxk∞. k 1nkxk∞=|xk|
|xk|p≤
n
P
i=1
|xi|p,kxk∞≤ kxkp.
∀x∈Cn,kxk∞≤ kxkp≤n1
pkxk∞.
k·k∞k·kp
p≥1q≥1k·kpk·kqRn.
lim
p→+∞n1
p= 1 n≥1 lim
p→+∞kxkp=kxk∞.